Using 3D-FEM for design of an axial flux seven-phase machine F. Locment, T. Henneron, E. Semail and F. Piriou L2EP, USTL Bâtiment P2, 59655 Villeneuve d'Ascq Cedex, France E-mail: [email protected]

Abstract ⎯For the experimental seven-phase machine studied in this paper, parameters necessary for the control such as inductances and electromotive force (EMF) are sensitive to harmonics. A conventional analytical method in which only the first harmonic is taken into account does not give good results. Besides, the machine is an axial-flux one and has two asymmetrical rotors: a 3D-FEM is then necessary. Comparisons between predeterminations and experimental results show sufficient accuracy to achieve a control model.

I.

INTRODUCTION

Axial flux permanent magnet (AFPM) synchronous three-phase machines are studied because their compactness and structure are interesting in embedded systems present in automotive [1], in electrical ship and in wind power [2]. However, if the main characteristics can be determined by analytical [3] or quasi-3D approaches [4] parasitic effects as cogging torque must be determined by 3D Finite Element Method. One drawback of these threephase machines is the difficulty to get sinusoidal EMF without complex repartition or shapes of the permanent magnets [4]. Technological constraints impose effectively simple windings. Besides, multiphase machines, intrinsically more reliable than three-phase ones, are also interesting for low voltage/high current speed drives also very common in embedded systems. However, leakage inductance and harmonics of magneto-motive forces have a strong impact on the determination of the PWM frequency chosen to limit observed parasitic currents [5]. Finite Element Method is also necessary to get precise results. One advantage of these machines is that torque ripples due to interactions between harmonics can be controlled to a low value by vector control even with non-sinusoidal EMF. In this paper a seven-phase AFPM machine is studied. For this machine, the usual drawback of non sinusoidal EMFs for three-phase machine does not exist. Experimental results are compared with 3D-FEM predeterminations. II.

3D – MODELING OF THE MACHINE

A. Numerical Model Due to the studied structure for the numerical model, the 3D-FEM is used in the magnetostatic case. We consider a domain D with a boundary Γ. In order to limit the number of unknowns, the scalar potential formulation (Ω-formulation) is classically used. Then, the equation to be solved can be written:

Np

div(µ(∑ K n i n − gradΩ)) = 0 (a)

(1)

n =1

B.n = 0 on Γ B , H × n = 0 on Γ H and Ω on Γ H

(b)

with Np the number of windings, Knin the source field and in the current associated to the winding n and µ the magnetic permeability. ΓB and ΓH are two complementary surfaces such as ΓB∪ΓH=Γ and ΓB∩ΓH=0. For a winding n, the vector Kn [6] of equation (1(a)) can be defined by:

curlK n = N n

(2)

with Nn the turn density vector flowing through the winding n. As the vector Nn is divergence free, there is an infinity of vectors Kn that verify the equation (2). To determine this vector, a tree technique is used. In Ω-formulation, the flux of the winding n can be written as follows [6]:

Φ n = ∫ B.K n dD

(3)

D

This expression will be used to determine the flux and the different inductances of the studied multi-phase machine. For the rotor motion, a slip surface is defined in the middle of the air gap, which requires a regular mesh [7]. The rotor displacement is modeled by a circular permutation of the unknowns, according to the mesh step. It may be noted that only rotor elements in contact with the slip surface are concerned by the permutation. At the level of the numerical algorithm, the unknown permutations are predetermined. So, the computation time and the storage memory do not increase when considering this movement. B. Presentation of the machine Figure 1 shows the studied machine. It has been modeled by a 3D-finite element method implemented in a software (CARMEL) developed in laboratory. The angular shift between the two identical six-pole rotors imposes the mesh of one sixth of the machine. It is composed of 791506 elements and 141937 nodes. To achieve a step of calculation it takes, on a Workstation bi-processor (80546k) XEON 3Ghz 2Go Ram, three minutes in linear mode and up to forty five minutes in saturated mode.

80

Measured EMF 3D-FEM EMF

60

MAGNETS ROTORS

40 EMF (V)

COILS

20 0 -20 -40 -60

STATOR

-80 0

0.01

0.02

0.03 0.04 Time (s)

Fig. 1. One sixth of the seven-phase studied machine.

0.05

0.06

Fig. 2. EMF of one phase of the seven-phase machine.

The soft-magnetic composite stator contains 42 slots (one slot per phase and per pole) in which a toroidal winding is inserted. The outside and inside diameters of the stator are 287 and 189 mm respectively. The two rotors are identical and the polarity of the magnets that are opposite each other is the same. There is a little angular shift of 360/84 degrees between the two rotors in order to reduce the cogging torque. Spatial repartition of the magnets allows the cancellation of the fifth harmonic of EMF. The overall thickness of the active parts is 123 mm and each airgap should measure 1mm. C. Characterization of the machine The EMFs have been measured at 300 rpm. We can see, in figure 2, that the waveforms are the same but that there is an error of proportionality that can be explained by the incertitude on airgap value. In fact, axial attraction forces, estimated to 6200N, and mechanical tolerances can induce an error of 150 µm on each airgap. A spectral analysis shows the same relative values of the first harmonic with the others harmonics (Table I.) and confirms the identity of the waveforms. TABLE I. HARMONIC BREAKDOWN OF EMF Order of harmonic Measured relative RMS values 3D-FEM relative RMS values

TABLE II. INDUCTANCES OF THE SEVEN-PHASE MACHINE

3

5

7

9

100% (53V)

19.3%

0.6%

6.6%

5.9%

100% (45V)

21.2%

0.3%

6.9%

6.3%

Relative error

(mH)

(%)

L

10.1

11.3

12

3.1

3.55

14.5

M13

-1.05

-1.14

8.6

M14

-5.3

-6.1

15

III.

CONCLUSION

The studied machine in this paper combines a typical 3D geometry and a sensitivity to harmonics because of its number of phases. So, a 3D Finite Element Method software has been used. The comparisons between numerical and experimental results show that it is possible to predetermine with sufficient accuracy parameters necessary for vector control such as EMF, inductance matrix and cogging torque. REFERENCES

[2]

Table II gives the values of inductances. Due to the symmetry of the machine, all self inductances L of the machine are equal and there are only three different mutual inductances (noted M12, M13, M14). The other values of mutual inductances can be obtained by permutation. Inductances are experimentally determined by imposing sinusoidal currents in one phase and measuring induced voltages in the other ones. Considering uncertainties on measurements and on characteristics of magnetic material, the values are close each other. In [5], the sensitivity of the time constants to the precision of the determination of these inductances has been developed. In the final paper, the comparison will be also achieved for the cogging torque of the machine.

3D-FEM value

(mH) M12

[1]

1

Measured value

[3]

[4] [5] [6] [7]

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